Question

Which is greater?
${\left(\frac{1}{2}\right)}^{\frac{1}{2}}$ or ${\left(\frac{2}{3}\right)}^{\frac{1}{3}}$

Answer 1

L.C.M of $2$ and $3$ is $6$.

$\therefore {\left(\frac{1}{2}\right)}^{\frac{1}{2}}=\sqrt[2]{2\frac{1}{2}}=\sqrt[2\times 3]{2\times 3{\left(\frac{1}{2}\right)}^3}=\sqrt[6]{6\frac{1}{8}}$

and ${\left(\frac{2}{3}\right)}^{\frac{1}{3}}=\sqrt[3]{3\frac{2}{3}}=\sqrt[3\times 2]{3\times 2{\left(\frac{2}{3}\right)}^2}=\sqrt[6]{6\frac{4}{9}}$

But $\frac{4}{9}>\frac{1}{8}$

$\therefore \sqrt[3]{3\frac{2}{3}}>\sqrt[2]{2\frac{1}{2}}$